joint work with Olga Brezhneva, Yuri Evtushenko
We present elementary proofs of a Karush-Kuhn-Tucker (KKT) theorem for an optimization problem with inequality constraints. We call the proofs "elementary" because they do not use advanced results of analysis. Our proofs of the KKT theorem use only basic results from Linear Algebra and a definition of differentiability. The simplicity of the proofs makes them particularly suitable for use in a first undergraduate course in optimization.
We will talk about a new family of cones called "amenable cones". We will show that feasibility problems over amenable cones admit error bounds even when constraint qualifications fail to hold. Therefore, amenable cones provide a natural framework for extending the so-called "Sturm's Error Bound". We will present many example of amenable cones and discuss the connection between the amenability condition and other properties such as niceness and facial exposedness. At the end, we will also discuss algorithmic implications of our results.
joint work with Roberto Andreani, Gabriel Haeser, Leornardo D. Secchin
Recently, the convergence of methods for nonlinear optimization problems has been analyzed using sequential optimality conditions. However, such conditions are not well suited for Mathematical Problems with Complementarity Constraints (MPCCs). Here we propose new sequential conditions for usual stationarity concepts in MPCC, namely weak, C- and M-stationarity, together with the associated weakest MPCC constraint qualifications. We also show that some algorithms reach AC-stationary points, extending their convergence results. The new results include the linear case, not previously covered.